# Properties

 Label 3630.2.a.bl Level $3630$ Weight $2$ Character orbit 3630.a Self dual yes Analytic conductor $28.986$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3630.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.9856959337$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 330) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + ( -2 - \beta ) q^{7} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + ( -2 - \beta ) q^{7} + q^{8} + q^{9} + q^{10} - q^{12} + ( -3 + 3 \beta ) q^{13} + ( -2 - \beta ) q^{14} - q^{15} + q^{16} + ( 4 - 4 \beta ) q^{17} + q^{18} + ( -3 + 3 \beta ) q^{19} + q^{20} + ( 2 + \beta ) q^{21} + ( -8 + \beta ) q^{23} - q^{24} + q^{25} + ( -3 + 3 \beta ) q^{26} - q^{27} + ( -2 - \beta ) q^{28} + ( 6 - 2 \beta ) q^{29} - q^{30} + ( -6 + 2 \beta ) q^{31} + q^{32} + ( 4 - 4 \beta ) q^{34} + ( -2 - \beta ) q^{35} + q^{36} + ( -5 - 3 \beta ) q^{37} + ( -3 + 3 \beta ) q^{38} + ( 3 - 3 \beta ) q^{39} + q^{40} + ( 2 + 5 \beta ) q^{41} + ( 2 + \beta ) q^{42} -2 \beta q^{43} + q^{45} + ( -8 + \beta ) q^{46} + ( 5 - 11 \beta ) q^{47} - q^{48} + ( -2 + 5 \beta ) q^{49} + q^{50} + ( -4 + 4 \beta ) q^{51} + ( -3 + 3 \beta ) q^{52} + ( -10 + 5 \beta ) q^{53} - q^{54} + ( -2 - \beta ) q^{56} + ( 3 - 3 \beta ) q^{57} + ( 6 - 2 \beta ) q^{58} + ( -5 + 7 \beta ) q^{59} - q^{60} + 6 \beta q^{61} + ( -6 + 2 \beta ) q^{62} + ( -2 - \beta ) q^{63} + q^{64} + ( -3 + 3 \beta ) q^{65} + ( -12 + 6 \beta ) q^{67} + ( 4 - 4 \beta ) q^{68} + ( 8 - \beta ) q^{69} + ( -2 - \beta ) q^{70} + ( 2 - 10 \beta ) q^{71} + q^{72} + 2 q^{73} + ( -5 - 3 \beta ) q^{74} - q^{75} + ( -3 + 3 \beta ) q^{76} + ( 3 - 3 \beta ) q^{78} + ( -6 - 4 \beta ) q^{79} + q^{80} + q^{81} + ( 2 + 5 \beta ) q^{82} + ( -4 + 4 \beta ) q^{83} + ( 2 + \beta ) q^{84} + ( 4 - 4 \beta ) q^{85} -2 \beta q^{86} + ( -6 + 2 \beta ) q^{87} + ( 6 - 9 \beta ) q^{89} + q^{90} + ( 3 - 6 \beta ) q^{91} + ( -8 + \beta ) q^{92} + ( 6 - 2 \beta ) q^{93} + ( 5 - 11 \beta ) q^{94} + ( -3 + 3 \beta ) q^{95} - q^{96} -2 q^{97} + ( -2 + 5 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 5q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 5q^{7} + 2q^{8} + 2q^{9} + 2q^{10} - 2q^{12} - 3q^{13} - 5q^{14} - 2q^{15} + 2q^{16} + 4q^{17} + 2q^{18} - 3q^{19} + 2q^{20} + 5q^{21} - 15q^{23} - 2q^{24} + 2q^{25} - 3q^{26} - 2q^{27} - 5q^{28} + 10q^{29} - 2q^{30} - 10q^{31} + 2q^{32} + 4q^{34} - 5q^{35} + 2q^{36} - 13q^{37} - 3q^{38} + 3q^{39} + 2q^{40} + 9q^{41} + 5q^{42} - 2q^{43} + 2q^{45} - 15q^{46} - q^{47} - 2q^{48} + q^{49} + 2q^{50} - 4q^{51} - 3q^{52} - 15q^{53} - 2q^{54} - 5q^{56} + 3q^{57} + 10q^{58} - 3q^{59} - 2q^{60} + 6q^{61} - 10q^{62} - 5q^{63} + 2q^{64} - 3q^{65} - 18q^{67} + 4q^{68} + 15q^{69} - 5q^{70} - 6q^{71} + 2q^{72} + 4q^{73} - 13q^{74} - 2q^{75} - 3q^{76} + 3q^{78} - 16q^{79} + 2q^{80} + 2q^{81} + 9q^{82} - 4q^{83} + 5q^{84} + 4q^{85} - 2q^{86} - 10q^{87} + 3q^{89} + 2q^{90} - 15q^{92} + 10q^{93} - q^{94} - 3q^{95} - 2q^{96} - 4q^{97} + q^{98} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.61803 −0.618034
1.00000 −1.00000 1.00000 1.00000 −1.00000 −3.61803 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 −1.38197 1.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$11$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3630.2.a.bl 2
11.b odd 2 1 3630.2.a.bf 2
11.c even 5 2 330.2.m.a 4
33.h odd 10 2 990.2.n.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.a 4 11.c even 5 2
990.2.n.h 4 33.h odd 10 2
3630.2.a.bf 2 11.b odd 2 1
3630.2.a.bl 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3630))$$:

 $$T_{7}^{2} + 5 T_{7} + 5$$ $$T_{13}^{2} + 3 T_{13} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$5 + 5 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$-9 + 3 T + T^{2}$$
$17$ $$-16 - 4 T + T^{2}$$
$19$ $$-9 + 3 T + T^{2}$$
$23$ $$55 + 15 T + T^{2}$$
$29$ $$20 - 10 T + T^{2}$$
$31$ $$20 + 10 T + T^{2}$$
$37$ $$31 + 13 T + T^{2}$$
$41$ $$-11 - 9 T + T^{2}$$
$43$ $$-4 + 2 T + T^{2}$$
$47$ $$-151 + T + T^{2}$$
$53$ $$25 + 15 T + T^{2}$$
$59$ $$-59 + 3 T + T^{2}$$
$61$ $$-36 - 6 T + T^{2}$$
$67$ $$36 + 18 T + T^{2}$$
$71$ $$-116 + 6 T + T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$44 + 16 T + T^{2}$$
$83$ $$-16 + 4 T + T^{2}$$
$89$ $$-99 - 3 T + T^{2}$$
$97$ $$( 2 + T )^{2}$$